Recognition of Flat Orbifolds and the Classification of Tilings in R 3
Discrete Comput Geom
Recognition of Flat Orbifolds and the Classification of Tilings in R3
O. Delgado-Friedrichs 0
0 Fakulta ̈t fu ̈r Mathematik, Universita ̈t Bielefeld , Postfach 100131, 33501 Bielefeld , Germany
We discuss criteria for an orbifold to carry a flat geometry, i.e., one which is modelled on euclidean geometry, and show how these lead to a practical flatness test in the three-dimensional case. This has immediate applications in combinatorial tiling theory, namely in the classification of three-dimensional periodic tilings up to equivariant equivalence, and in the material sciences.
According to Thurston [Th2], a d-dimensional orbifold is a space which is (compatibly)
locally modelled after Rd modulo finite group actions. Its singular locus is the set of points
on which the group action is non-trivial. The underlying space is stratified according to
the essentially different actions which occur. A cellular subdivision of the underlying
space is admissible with respect to the orbifold structure if each open cell is contained
in some stratum. As with manifolds, it is often useful to describe an orbifold in terms of
the combinatorial structure of an admissible cellular subdivision, e.g., a triangulation,
together with appropriate information on the groups associated to each cell.
The question then arises whether a given orbifold is good, i.e., is covered—in the
sense of an orbifold covering—by some simply connected manifold and, if so, whether
there is a compatible geometry on that manifold.
In combinatorial tiling theory, we represent periodic tilings by special triangulations
of the orbifolds derived as orbit spaces of their respective symmetry groups. These
special orbifold triangulations, or, more precisely, the particular combinatorial descriptions
which turned out to be useful, are called Delaney symbols and were introduced by Dress
in the mid-1980s [Dr1], [Dr2]. Based on Delaney symbols, systematic methods for the
classification, generation and visualization of tilings in two and three dimensions have
been developed and implemented [DH1], [De1], [Hu2], [BH], [DH3]. These have been
successfully applied in the solutions of numerous classification tasks, as, e.g., [DH2],
[DHZ], [Hu3], [DHM], [DH5]. The 1-skeleton of a periodic three-dimensional tiling can
be interpreted as the atom-bond-network of a potential crystal. We have found that many
known crystal networks—and of course many new and possibly interesting structures—
can be obtained in this way from relatively simple tilings [DDHC], [DH5]. Thus practical
enumeration methods for three-dimensional tilings have direct implications in the
The basic idea of the Delaney symbol approach seems to be a very natural one, as it
has been reinvented several times. Most notably, there is the theory of coloured graphs,
crystallizations and GEMs (graph-encoded manifolds) as established by Pezzana and
developed further by Ferri, Gagliardi, Lins and others (see [LM] and [Li]). Delaney
symbols can be regarded as a generalization of GEMs to the orbifold case.
A central problem in combinatorial tiling theory is to determine whether a given
wellformed symbol gives rise to a tiling in the desired geometry. In two dimensions this can
be solved by simple criteria based on the Euler characteristic and the well-known
classification of good two-dimensional orbifolds (see [Vi]). The three-dimensional problem is
much harder, even if we restrict our attention to tilings of euclidean space. In effect, we
have to decide whether a given 3-orbifold represented by a Delaney symbol is covered by
R3 in such a way that all homeomorphisms of R3 which are annihilated by the covering
projection are euclidean isometries.
The main theoretical result of this paper is that, indeed, this can be done, so, at
least in principle, tilings of three-dimensional space can be generated effectively (see
Theorem 15). In a second part, we consider strategies to obtain an algorithm which is
fast enough to be actually useful. More details on this, including complexity results, will
be given in a forthcoming paper.
Many commonly used representations of orbifolds can be easily converted to
admissible triangulations, and these in turn can be converted to Delaney symbols. Thus, our
approach to the flatness problem is of relevance not only to tiling theory.
In the following section we introduce definitions and notation. We present and discuss
a general scheme for flatness tests in Section 3 and its application to the three-dimensional
case in Sections 4 and 5. In Sections 6–8 we discuss a practical approach for dimension 3.
Finally, in Sections 9, we illustrate our methods by an example.
Delaney symbols have been discussed and applied in many papers. Here we give a short
introduction and state some basic facts for reference.
Consider a simply connected d-manifold X (without boundary) together with a regular
CW-filtration X D fX0; : : : Xd g, ; D X¡1 ½ X0 ½ X1 ½ ¢ ¢ ¢ ½ Xd D X . Such a
CWfiltration is called a tiling of X . Let 0 be a group acting on X by self-homeomorphisms and
respecting the filtration, i.e., satisfying ° .Xi / D Xi for all ° 2 0 and all i D 0; : : : ; d.
There are obvious induced actions of 0 on the sets of cells of any given dimension and
on the set of inclusion chains of cells. In particular, there is an action of 0 on the set of
flags (maximal chains)
F D .F0; : : : ; Fd / 2 Y ¼0.Xi ¡ Xi¡1/ j Fi¡1 µ Fi for i D 1; : : : ; d :
We can, on the other hand, use the fact that for each flag F D fF0; : : : ; Fd g and each
i 2 f0; : : : ; dg there is a unique flag F 0 D fF00; : : : ; Fd0 g with Fj D Fj0 if and only j 6D i ,
to define a second group action on F . Let
6 D 6I :D ¾0; : : : ; ¾d j ¾02 D ¢ ¢ ¢ D ¾d2 D 1®
denote the free Coxeter group with index set I :D f0; : : : ; dg. As a shortcut in notation,
we usually skip the braces when giving I explicitly, thus 60;1;2 :D 6f0;1;2g.
A right action of 6 on F is obtained by defining for all F 2 F and all i 2 I the image
F ¾i as the unique flag which differs from F in exactly its i -dimensional component.
The actions of 0 and 6 on F commute, so the 6-action carries over to the orbit space
D D DX ;0 :D 0nF :D f0 F j F 2 F g.
In the case where 0 acts freely on X , the CW-complex X is determined uniquely
up to compatible homeomorphisms by the abstract 6-set D. If 0 does not act freely,
additional information specifying the stabilizers of each cell is needed. It turns out to be
sufficient to provide a map
M D MX ;0: D £ I £ I ! N;
.D; i; j / 7!
mi; j .D/;
where mi; j .D/ :D ri; j .C / :D inffr > 0 j C .¾i ¾j /r D C g for some C 2 D (note that
the value of mi; j .D/ does not depend on the choice of C ). The quotient vi; j .D/ :D
mi; j .D/=ri; j .D/ is sometimes called a branching limit or branching number. It is easy
to see that a branching limit v > 1 indicates a rotational symmetry of order v.
The pair .DX ;0; MX ;0/ is called the Delaney symbol of the 0-equivariant
CWfiltration X on X .
From now on, we are mostly concerned with generalized formal Delaney symbols—
D-symbols for short—by which we mean pairs .D; M / where D is a transitive 6I -set
for some finite index set I ½ N and M is a map from D £ I £ I into N such that the
following obvious conditions hold for all D 2 D and all i; j 2 I :
(DS0) mi;i .D/ D 1.
(DS1) mi; j .D/ D mi; j .D¾i / D m j;i .D/.
(DS2) D.¾i ¾j /mi;j .D/ D D.
We regard D-symbols as a generalized discrete analogue of orbifolds. A morphism (or,
to stress the analogy, a covering) between D-symbols .D; M / and .D0; M 0/ is a map
': D ! D0 satisfying '.D¾i / D .'.D//¾i and mi; j .'.D// D mi; j .D/ for all D 2 D
and all i; j 2 I . A functor TOP (for topological realization) from the category of
Dsymbols back into the category of topological spaces is obtained by associating to each
element a copy of the .#I ¡ 1/-dimensional standard simplex with facets labelled by
the elements of I and identifying the i -labelled facets of two simplices whenever the
corresponding D-symbol elements are interchanged by ¾i . The identification is to be
done in such a way that the remaining labels match, so TOP.D/ is uniquely determined
up to homeomorphisms. Accordingly, the dimension of a D-symbol is defined as one
less than the cardinality of its index set I . The topological realization of a D-symbol
morphism is defined as the direct product with the identity map on the standard simplex
modulo identifications. It is easy to see that the result is a well-defined continuous map.
The analogue to universal covers is provided by what we call the universal preimage.
A D-symbol .De; Me/ is called a universal preimage of a D-symbol .D; M / with respect to
base points De 2 De and D 2 D if there is a morphism ': .De; Me/ ! .D; M / which maps
De to D such that for any morphism Ã : .D0; M 0/ ! .D; M / and any fixed D0 2 Ã ¡1.D/
there is a uniqe '0: .De; Me/ ! .D0; M 0/ with '0.De/ D D0 and ' D Ã ± '0.
By the usual arguments, a universal preimage must—up to isomorphism—be unique
and independent of the chosen base points. Its existence is established by the following
construction: for some fixed D 2 D, consider the subgroup
3 D 3D.D; M / :D h¿ .¾i ¾j /mi;j .D¿/¿ ¡1 j ¿ 2 6I i; j 2 I i · 6:
Now, let De :D 3n6 :D f3¿ j ¿ 2 6g, the set of cosets of 3 in 6, which is a transitive
6-set via the usual action .3¿ /½ D 3.¿½/, and set
': 3n6 ! D;
3¿ 7! D¿:
Obviously, 3 is contained in the 6-stabilizer of D, so by standard arguments, ' is a
welldefined 6-map. We define Me by mi; j .3¿ / :D mi; j .D¿ /, so ' fulfils the requirement
for a morphism of D-symbols. To show that .De; Me/ is indeed a D-symbol, we note
that (DS0) and (DS1) are immediate, while for each 3¿ 2 De and each pair i; j 2 I ,
the group element ¿ .¾i ¾j /mi;j .D¿/¿ ¡1 is, by definition, in the stabilizer of 3 2 De, so
.¾i ¾j /mi;j .D¿/ D .¾i ¾j /mi;j .3¿/ is in the stabilizer of 3¿ , which proves (DS2). Universality
follows from the fact that 3 is indeed defined such as to be contained in the 6-stabilizer
of any preimage of D by a D-symbol morphism.
As with orbifolds, the universal cover, or, in this case, the topological realization of the
universal preimage, may or may not be a manifold. For orbifolds, even if the underlying
space is a manifold, there may be non-trivial local group actions. In a D-symbol we have
similar phenomena, which we subsume under the following definitions:
Definition 1. A D-symbol .D; M / is called unwound if, for each D 2 D and all
i; j 2 I , we have vi; j .D/ D 1 or, equivalently, D.¾i ¾j /r 6D D for 0 < r < mi; j .D/. It
is called loopless if D¾i is always different from D. A D-symbol is called semi-good if
its universal cover is unwound.
The universal cover is always loopless. In fact, it is easy to see that it is always
oriented, meaning that there is a function d: De ! f¡1; C1g with d.D¾i / never equal
Definition 2. A D-symbol .D; M / with universal preimage .De; Me/ is called good if it
is semi-good and TOP.De; Me/ is a manifold.
The smallest example of a non-semi-good D-symbol is given as follows: let D :D
f A; Bg and I :D f0; 1; 2g, A¾0 D A¾1 D A, B¾0 D B¾1 D B, A¾2 D B, m0;2. A/ D
m1;2. A/ D 2, m0;1. A/ D 1 and m0;1.B/ D k > 1. The universal preimage of .D; M /
(which has four elements) corresponds to the so-called teardrop orbifold, a 2-sphere
with a single singular “cone” point.
The fundamental group of a D-symbol .D; M / is defined as the group ¼1.D; M / of all
automorphisms ® of the universal preimage satisfying ' ± ® D ', where ': .De; Me/ !
.D; M / is the canonical projection, as above. It can be shown that ¼1.D; M / is isomorphic
to 6D=3D.D; M /, where 6D is the 6-stabilizer of D 2 D. Balke and Valverde have
described a finite presentation of this group for finite D in [BV], which is a good starting
point for practical calculations.
We are now in a position to state the flatness problem in the context of D-symbols:
Definition 3. A d-dimensional D-symbol .D; M / with universal preimage .De; Me/ and
fundamental group ¼1.D; M / is called flat or euclidean if it is semi-good and there is a
®: TOP.De; Me/ ! Rd
such that for all ½ 2 ¼1.D; M / the map ® ± TOP.½/ ± ®¡1 is an isometry with respect to
the usual euclidean metric on Rd .
Of course, we can define hyperbolic or spherical D-symbols in a similar way. An
obvious necessary condition for a D-symbol to be good is that it locally “unfolds”
correctly. This can be made precise by introducing appropriate substructures. For .D; M /
a D-symbol with index set I , D 2 D an arbitrary element and J µ I , we define the
Dsubsymbol generated by D and J to be the 6J -set D.D;J/ :D D6J , where 6J is regarded
as a subgroup of 6I in the obvious way, together with the map M.D;J/ :D M jD6J £J£J .
Now a d-dimensional D-symbol is called locally flat if all its .d ¡ 1/-dimensional
subsymbols are spherical. Every good D-symbol must be locally flat.
Fortunately, there are simple effective criteria for determining whether a given
twodimensional D-symbol is spherical, euclidean or hyperbolic [DH1], so, in particular,
local flatness is easy to check in three-dimensional D-symbols. Namely, let .D; M / be
a D-symbol with index set I D fi; j; lg. We define a function k: D ! Q by
1 1 1
k.D/ :D ki; j;l .D/ :D mi; j .D/ C mi;l .D/ C m j;l .D/ ¡ 1
and set K .D; M / :D Ki; j;l .D; M / :D PD2D ki; j;l .D/. The number K .D; M / is called
the curvature of the D-symbol .D; M /. It is closely related to the orbifold Euler
characteristic in the sense of [Th2], which for manifolds coincides with the usual Euler
characteristic. It is thus not surprising that the sign of K .D; M / distinguishes the
curvature of the metric for a realization of .D; M /. More precisely, we have:
Theorem 4. A two-dimensional D-symbol .D; M / is
² hyperbolic, iff K .D; M / < 0 holds,
² euclidean, iff K .D; M / D 0 holds,
² spherical, iff K .D; M / > 0 holds and, additionally, for all i; j 2 I and all D 2 D
vi; j .D/ ¢ K .D; M /
is a natural number and
² non-semi-good in all other cases.
If H is a group of automorphisms of a D-symbol .D; M /, then we denote by .D; M /=H
the unique image .D0; M 0/ of .D; M / with D0 D D=H . In analogy to the topological
setting, we find that for each subgroup H of ¼1.D; M /, we have a D-symbol morphism from
.De; Me/=H onto .D; M /, which is an isomorphism exactly in the case H D ¼1.D; M /.
3. Flatness in General Dimension
In this section we discuss flatness criteria for finite D-symbols of arbitrary dimensions.
The topological realization of a finite D-symbol is compact, and there are only finitely
many compact flat orbifolds in each dimension, as was shown by Bieberbach in his
famous work [Bi]. Actually, Bieberbach proved that there are finitely many geometric types
of crystallographic groups in each dimension. This result translates directly to orbifolds,
see [Th3]. It would suffice therefore to compare the given symbol against each of these
finitely many cases. Of course, this procedure is not the most practical in dimension 2,
extremely tedious if possible at all in dimension 3 and hopeless in higher dimensions.
Fortunately, every flat d-orbifold is finitely covered by a d-dimensional torus Rd =Zd .
This follows from another one of Bieberbach’s famous theorems, stating that each
ddimensional crystallographic space group G (i.e., each co-compact, properly
discontinuous group of euclidean isometries in dimension d) has a subgroup of finite index which
is generated by d linearly independent translations. Moreover, the subgroup T .G/ of all
translations in a given group is of this type and is a normal and maximal abelian subgroup.
Based on this fact, we propose the following preliminary sketch of a flatness test:
Algorithm 5. Given a finite locally flat d-dimensional D-symbol .D; M /.
1. Compute a presentation for G :D ¼1.D; M /.
2. For each positive integer k in order, enumerate all subgroups of G of index k.
3. Pick the first of these subgroups which is isomorphic to Zd and call it T (if none
such exists, the D-symbol is not flat).
4. Compute the image .D0; M 0/ :D .De; Me/= T .
5. Determine whether .D0; M 0/ is unwound and its topological realization is a d-torus.
6. If not, continue at Step 3, this time picking the next unused subgroup.
If the maximal possible index of the translation subgroup is known, this procedure is
finite, provided, of course, that each individual step can be carried out in finite time. If
the test in Step 5 is negative for all subgroups picked in Step 3, then the given D-symbol
is not flat, so in the general case, we only check a necessary condition for flatness. We
will see later that in dimension 3 this condition is already sufficient.
Of course, in a certain sense, Step 3 cannot be carried out at all, because the
isomorphism problem for finitely presented groups is undecidable. The following result
Lemma 6. If G is a space group of dimension d and N is a finite index subgroup of G,
then N is contained in T .G/ if and only if N =[N ; N ] is isomorphic to Zd . In particular,
if G=[G; G] »D Zd holds, then we have G D T .G/.
Proof. If N has finite index in G and is contained in T .G/, then N is a subgroup of
finite index in T .G/, so it must of course be isomorphic to Zd .
To show the other implication, first note that any subgroup of finite index of a space
group is again a space group, so we may restrict our attention to the case N D G.
If G=[G; G] has rank d, then .T .G/¢[G; G]/=[G; G], being a subgroup of finite index
in G=[G; G], also has rank d. However, then .T .G/ ¢ [G; G]/=[G; G] »D T .G/=.T .G/ \
[G; G]/ and T .G/ both have rank d, which forces T .G/ \ [G; G] to be trivial.
Now if g is an arbitrary element of G, and h is an arbitrary element of T .G/, then
[g; h] is in T .G/, since T .G/ is a normal subgroup of G. By the above, then, we must
have [g; h] D 1, so every element of G commutes with every element of T .G/. In
particular, for any g 2 G, the subgroup of G generated by T .G/ and g, is abelian.
However, since T .G/ is a maximal abelian subgroup of G, this implies G D T .G/.
Using this, we can replace Step 3 of Algorithm 5 by Step 30 below and, moreover,
dispose of Step 6, because if the original D-symbol is flat, then, using the homeomorphism
® from Definition 3, ® ± TOP.T / ± ®¡1 must be a translation subgroup of rank d in the
d-dimensional crystallographic space group ® ± TOP.G/ ± ®¡1.
Algorithm 7. Given a locally flat d-dimensional D-symbol .D; M /.
1. Compute a presentation for G :D ¼1.D; M /.
2. For each positive integer k in order, enumerate all subgroups of G of index k.
30. Pick the first of these subgroups, call it T , such that T =[T ; T ] is isomorphic to Zd .
4. Compute the image .D0; M 0/ :D .De; Me/= T .
5. Determine whether .D0; M 0/ is unwound and its topological realization is a
In other words, we have:
Theorem 8. If .D; M / is a flat d-dimensional D-symbol with universal cover .De; Me/
and fundamental group G, then there is a subgroup T of finite index in G such that
T =[T ; T ] is isomorphic to Zd and for any such subgroup the image .De; Me/= T is unwound
and its topological realization is homeomorphic to a d-dimensional torus.
Now, as mentioned above, Step 1 can be performed by computing the
presentation of Balke and Valverde, which utilizes the identity ¼1.D; M / »D 6D=3D.D; M /,
a Reidemeister–Schreier construction for 6D and the fact that 3D.D; M / is normal in
6D. The last statement implies that a presentation for 6D=3D is obtained from one for
6D simply by appending a list of generators for 3D to the list of relators.
Given a finite presentation for ¼ :D ¼1.D; M /, all subgroups up to some finite
index m can be found by the standard method of systematically generating all possible
actions of ¼ on finite sets of sizes up to m, followed again by a Reidemeister–Schreier
construction to obtain finite presentations (see [Ne]). If N is such a subgroup, the rank
and the torsion elements of N =[N ; N ] can be determined by another standard method,
namely by diagonalizing its relator matrix over Z, after which the comparison against
Zd in Step 30 becomes trivial.
For Step 4, assume that a subgroup H 0 of ¼ is given via a permutation representation
of ¼ on the set of cosets ¼=H 0. Let H be its preimage with respect to the canonical
projection 6D ! 6D=3D D» ¼ . To simplify notation, we may identify ¼=H 0 with
6D=H and De=H 0 with De=H , where H now acts in the usual way as a subgroup of 6.
The following lemma tells us how to obtain De=H from D and 6D=H :
Lemma 9. Let G be a group with e its neutral element, let ': S ! T be a G-map
between transitive G-sets S and T and let s 2 S be arbitrary but fix and set t :D '.s/ 2 T .
Let r : T ! G be such as to satisfy r .t / D e and t ¢ r .t 0/ D t 0 for each t 0 2 T . Finally,
let H be a subgroup of Gt which contains Gs .
Then the G-set S=H is isomorphic to T £ Gt =H via
where G acts on T £ Gt =H by
for all t 0 2 T , a 2 Gt and g 2 G.
S=H ! T £ Gt =H;
s H g 7! .tg; H g ¢ r .tg/¡1/;
.t 0; H a/g :D .t 0g; H a ¢ r .t 0/ ¢ g ¢ r .t 0g/¡1/
Proof. It is easy to see that the above defines a G-action on T £Gt =H . Since .t 0; H a/ D
.t; H /a ¢ r .t 0/ for all t 0 2 T and a 2 Gt , the action is transitive. It suffices to show that
the stabilizers of .t; H / and s H 2 S=H are the same, i.e., that the stabilizer of .t; H / is
just H . Obviously for each h 2 H , we have .t; H /h D .t h; H h/ D .t; H / because of
H · Gt . If on the other hand, .t; H /g D .t; H / holds for some g 2 G, we have t D tg,
thus g 2 Gt , which in turn gives r .tg/ D e, thus H g D H and finally g 2 H .
4. Recognizing the 3-Torus
There is no known general algorithm to test whether two three-dimensional manifolds
are isomorphic. In spite of this, manifolds with sufficiently simple combinatorial
representations have been successfully recognized and catalogued, as for example in [Li] and
Based on work by Haken [Ha2], it has been shown that for a special class of
3manifolds, now called Haken manifolds, the isomorphism problem can be solved
algorithmically (see [He2]). A Haken manifold is a compact, oriented, irreducible 3-manifold
which contains an incompressible surface of positive genus. Essentially, a 3-manifold M
is called irreducible if every 2-sphere properly embedded in M bounds a 3-cell and an
oriented surface S of positive genus properly embedded in M is called incompressible
if every loop in S which bounds a disk in M bounds a disk in S.
Haken also describes a method to split a given PL-manifold M into irreducible pieces
[Ha1], i.e., to compute a collection of PL-manifolds M1; : : : ; Mk such that each Mi is
either irreducible or isomorphic to S2 £ S1 (the only reducible 3-manifold which is no
non-trivial connected sum) and such that M is just the connected sum M1# ¢ ¢ ¢ #Mk . The
connected sum of two manifolds is obtained by removing a cell from the interior of
each and gluing together the resulting manifolds along the boundaries of these cells. It
has been shown that for oriented manifolds the result does not depend on the particular
gluing and, moreover, that every compact oriented 3-manifold different from the
3sphere is uniquely decomposable with respect to connected sums into a collection of
so-called prime manifolds. A manifold different from S3 is called prime unless it is a
connected sum of two manifolds both different from S3, i.e., if it is either incompressible
or isomorphic to S2 £ S1 (see [He1]).
Unfortunately, there can be copies of S3 among the pieces obtained by Haken’s
method, i.e., the collection of prime manifolds obtained may not be minimal. As
obviously the 3-sphere does not contain an incompressible surface, Haken’s method is not
applicable to it. An algorithm to recognize the 3-sphere based on so-called almost
normal surfaces was recently presented by Rubinstein [Ru] and elaborated on by Thompson
[Th1]. We refer to it as the Rubinstein–Thompson method.
Theorem 10. There is an algorithm to check whether a given PL-manifold is
homeomorphic to the 3-torus T 3 :D S1 £ S1 £ S1.
Proof. Let M be a PL-manifold given in a suitable way, e.g., as a triangulation. Denote
by H the first integral homology group of M. If M is the 3-torus, then H must be
isomorphic to Z3 as a group. This can be checked in finite time, and in the following, we
assume that it is the case.
Using Haken’s method, split M into a collection of prime manifolds M1; : : : ; Mk with
homology groups H1; : : : ; Hk such that M is isomorphic to M1# ¢ ¢ ¢ #Mk . Necessarily
then, we have H D H1 £ ¢ ¢ ¢ £ Hk . The 3-torus is prime, so M is isomorphic to T 3
if and only if exactly one of the pieces is, while all other pieces are isomorphic to S3.
We may assume without loss of generality that H1 D Z3 and Hi D 1 for i D 2; : : : ; k
Every oriented 3-manifold with infinite first homology contains an incompressible
surface of positive genus (see [He1]), thus, as H1 is irreducible by construction, it must
be Haken. We can therefore apply the isomorphism test for Haken manifolds.
On each of the remaining pieces, the Rubinstein–Thompson test can be applied to
check whether all of them are isomorphic to S3.
5. Flatness and D-Symbol Morphisms
Theorem 8 provides a necessary flatness condition for D-symbols of arbitrary
dimensions. Roughly, it says that a cover with a certain property must exist and that the cover
must be flat. In general, any cover of a flat D-symbol is flat, because its fundamental
group will be a subgroup of the original symbol’s. If the reverse was true as well, this
would imply the sufficiency of the condition from Theorem 8.
In dimensions 2 and 3 this is indeed the case. In effect, every periodic tiling of
R2 or R3 can be realized with its full combinatorial symmetry. In dimension 2 this
even holds for spherical and hyperbolic tilings and is an easy consequence of the
geometric classification theorem for two-dimensional D-symbols [DH1], [Hu1] (see
In dimension 3 the result follows from an important theorem in geometric topology
Theorem 11 [MS]. Let M be a 3-manifold with a geometric structure modelled on one
of H 2 £ R, S]L2R, Nil, E3 or Sol. Then, up to conjugacy, any smooth finite group action
on M preserves the geometric structure on M .
By standard arguments, the conclusion also holds for group actions which fix a
(pseudo-)triangulation of the given manifold, so in particular, it can be applied to the
case where G is a group of D-symbol automorphisms acting on D, TOP.D/ allowing
for a geometric structure of one of the five listed types. The theorem then implies that
the given metric can be adjusted via a self-homeomorphism of TOP.D/ to one on which
TOP.G/ acts by isometries.
Theorem 12. If .D; M / and .D0; M 0/ are finite three-dimensional D-symbols and there
is a covering ': .D; M / ! .D0; M 0/, then .D; M / is flat if and only if .D0; M 0/ is flat.
Proof. If .D0; M 0/ is flat, then trivially this holds for .D; M /. In the following, we
assume that .D; M / is flat and show that .D0; M 0/ is as well.
Let G :D ¼1.D; M / and G0 :D ¼1.D0; M 0/ be the fundamental groups of .D; M /
and .D0; M 0/, respectively. We may assume without loss of generality that .D; M / is
unwound, thus TOP.D/ is a manifold, and that G is normal in G0. Indeed, any finite
cover of .D; M / will be flat. By Theorem 8, .D; M / has an unwound finite cover, so we
assume that .D; M / is unwound. If G is not normal in G0, then we replace .D; M / by
the finite cover corresponding to the core of G in G0.
Because G is normal in G0, the action of G0 on .De; Me/ induces an action on .D; M /.
Now since .D; M / is flat, TOP.D/ is homeomorphic to the orbit space of E3 by an
isometric G-action. In effect, G0 acts on the flat manifold TOP.D/ by homeomorphisms,
fixing the pseudo-triangulation induced by the 6-set structure. The finite index subgroup
G of G0 acts trivially, so the G0-action is effectively only a finite one. By Theorem 11,
we may assume that G0 preserves the induced flat metric on TOP.D/.
Lifting to the universal cover, we finally obtain the desired isometric action of G0
Theorem 12 has two important consequences. Firstly, it tells us that in dimension 3
the necessary condition from Theorem 8 is also sufficient. Because of its importance,
we state this as a separate theorem.
Theorem 13. Let .D; M / be a three-dimensional D-symbol with universal cover .De; Me/
and fundamental group G.
If .D; M / is flat, then there is a subgroup T of index at most 48 in G such that T =[T ; T ]
is isomorphic to Zd .
If, on the other hand, T is such a subgroup, then .D; M / is flat if and only if the
image .De; Me/= T is unwound and its topological realization is homeomorphic to a
Corollary 14. There is an algorithm to check whether a given three-dimensional
Dsymbol is flat.
Proof. This follows directly from Theorems 10 and 13.
Our main result follows:
Theorem 15. For each natural number n, the collection of all (isomorphism classes
of) D-symbols of size not greater than n corresponding to periodic tilings of
threedimensional euclidean space can be computed in finite time.
Proof. Obviously, there are up to isomorphism only finitely many 6-sets of each given
size. As the branching limits in flat D-symbols are subject to the crystallographic
restriction, i.e., can only be 1, 2, 3, 4 or 6, there are only finitely many D-symbols of each
given size which have to be checked for flatness.
Strictly speaking, Theorem 15 does not say that we can construct an actual
representative of each equivalence class of three-dimensional euclidean tilings. All we know is
that such a tiling exists. The main reason for this is that we have used Theorem 11 as a
“black box”, a mere existence result. However, the proof given by Meeks and Scott
actually is constructive. It does not seem unreasonable therefore to believe that Algorithm 7
can be extended to construct a tiling for each D-symbol found to be flat. A detailed
description of such an extended algorithm however is beyond the scope of this article.
In applications so far, heuristic methods have been used which were successful in most,
but not all, cases.
Secondly, it suffices to test flatness for only one representative of each class of
Dsymbols which are connected by some chain of coverings. We call a D-symbol minimal
if there is no non-trivial D-symbol morphism from it onto any other D-symbol. It can
be shown that to each D-symbol .D; M / there is a unique minimal image, called the
universal image, such that every other image of .D; M / also maps onto it. Therefore,
in classifications of three-dimensional tilings, we usually restrict ourselves to mimimal
D-symbols. These, if flat, represent tilings which are maximally symmetric in their
respective topological families.
6. Speeding up the Subgroup Search
The complexity of subgroup enumeration, as required in Step 2 of Algorithm 7, is
estimated to grow at least exponentially with the index. In dimension 3 already, this
index can be as large as 48. We discuss two strategies to avoid enumerating large index
The first strategy is applicable whenever sufficiently much is known on the structure
of possible point groups. Every euclidean isometry can be described as the product of
an orthogonal linear map and a translation. The map which takes each isometry to its
linear component is a group homomorphism. For a given crystallographic space group,
the image is called its point group and (obviously) is isomorphic to the quotient of the
original group by its translation subgroup. The possible point groups are exactly those
finite subgroups of the full orthogonal group which fulfil the so-called crystallographic
restriction, i.e, do not contain non-trivial rotations of orders other than 2, 3, 4 or 6. In
dimension 3 there are, up to conjugation in O3.R/, precisely 23 such groups, all of which
are isomorphic to some group of the form H , H £ C2 or H £ C2 £ C2, where H is a
subgroup of S4 and C2 is the group with two elements.
All candidates for translation subgroups of a given D-symbol fundamental group are
thus found by enumerating kernels of homomorphisms into S4 and intersecting these
with up to two subgroups of index 2. To enumerate the kernels, all conjugacy classes of
subgroups of index up to 4 are constructed. For each of these subgroups its core, i.e.,
the largest normal subgroup of the original group contained in it, is determined, again
by intersection methods [NF], [De2].
The second strategy is based on geometric features of the given D-symbol and its
topological realization. It works by successively computing subgroups of small index and
their corresponding covers and in each step choosing exactly one which is in some sense
simpler than the previous symbol. In [De2] it is shown that for each three-dimensional
euclidean D-symbol a subgroup of the translation group can be found by successively
constructing covers of index 3 at most twice and of index 2 at most five times.
As a first step, it is easy to see that, in any dimension, a D-symbol is oriented if and only
if the symmetry group associated to it acts in an orientation preserving way. For a
nonoriented D-symbol, there is a canonical oriented two-sheeted cover which corresponds to
the orientation-preserving subgroup. Succeeding steps proceed by “eliminating” certain
types of singular points from the corresponding orbifold until the resulting cover is a
manifold and then its first integral homology is used as an invariant. It is conceivable
that similar methods will be useful for other metrics or higher dimensions.
In practice, combinations of the two strategies have been most successful. In
dimension 3 for example, point groups of the form H £C2 £C2 are never orientation preserving.
In [DH3] we describe certain invariants which distinguish the 219 crystallographic space
groups in dimension 3. These can be used to avoid the subgroup search completely in
some cases, while in others, advance knowledge of the (potential) point group will allow
a restricted search.
7. The 3-Torus Recognition Problem in Practice
The key to practical manifold recognition is to simplify a given representation as far as
possible, thus obtaining either a manifold homeomorphic to the original one, or else a
collection of connected summands of the original manifold. If the resulting representations
are small enough, they can be looked up in a catalogue. Otherwise, more sophisticated
methods can be applied. Algorithms based on the enumeration of embedded surfaces
as referred to in Section 4 tend to have extremely high computational complexity. Thus
almost every effort to reduce the size of their input data should be worthwhile.
The strategy sketched below for the torus-recognition problem has, in spite of its
obvious limitations, proved successful in many applications [De2], [DH4], [DDHC],
[DH5]. In this and the following section we only consider unwound D-symbols, so we
restrict our attention to the underlying 6-sets. Moreover, we always assume that these
are three-dimensional, oriented and locally flat, thus their topological realizations are
oriented three-dimensional manifolds (see [LM]).
Algorithm 16. Let D be an oriented locally flat 6-set with fundamental group G such
that G=[G; G] »D Z3. The algorithm searches for criteria telling whether TOP.D/ is
homeomorphic to the three-dimensional torus T 3 D S1 £ S1 £ S1. Possible outcomes:
“yes”, “no” or “unsure” (the latter meaning that no criterion was found which would
allow a decision).
1. Try to prove inequality by group-theoretic invariants. In detail:
(a) Compute the number k of conjugacy classes of subgroups of G with index up
to 4. If k is different from 56 (the corresponding number for Z3), return “no”.
(b) Determine whether the (abelian) quotient [G; G]=[G; [G; G]] is trivial.
Otherwise, return “no”. We use a method described in [Ni] to compute this quotient.
2. Apply a simplification procedure to obtain either
(a) a simpler 6-set D0 such that TOP.D0/ is homeomorphic to TOP.D/ or
(b) a proof that TOP.D/ is not a 3-torus (return “no” in this case).
3. Check whether D is contained (up to an isomorphism) in a short list of simple
representations of the 3-torus. If so, return “yes”, otherwise, return “unsure”.
We start this section by describing a rather general type of operation on 6-sets due to
Dress (see [Fr]). Let D be a finite 6-set with index set I . For some i 2 I , let A be a
¾i -invariant subset of D, thus D¾i 2 A for all D 2 A. Define a new 6-set D0 as the set
D ¡ A together with the 6-action ±: D0 £ 6 ! D0 given by D ± ¾j :D D¾j .¾i ¾j /r
where r :D minfn 2 N0 j D¾j .¾i ¾j /n 2 D0g, for all D 2 D0 and j 2 I . It is not hard to
show that r is always well-defined and that . D ± ¾j / ± ¾j D D holds for all relevant D
and j , so ± is indeed a 6-action. We call the operation of removing a ¾i -invariant subset
A and redefining the 6-action as above an i -reduction and denote the reduced 6-set D0
by Dni A. We sometimes abuse this notation to refer to the specific reduction itself.
It is easy to see that for disjoint i -invariant subsets A and B, we have .Dni A/ni B D
Dni .A [ B/. To study the behaviour of i -reductions in general, we consider elementary
i -reductions, where A is a single ¾i -orbit, thus A D f D; D¾i g for some D 2 D. In
the following we assume that 6 is loopless, so every ¾i -orbit has exactly two elements.
Define J :D f j 2 I j D¾j D D¾i g. There are essentially three cases:
1. In the special case J D I , if D is transitive, TOP.D/ is homeomorphic to an
n-sphere and the result of the i -reduction is empty. Otherwise, a connected
component of TOP.D/ homeomorphic to an n-sphere is removed.
2. If D6I ¡J and D¾i 6I ¡J are disjoint, then the effect of the reduction on the
topological realization of D is that of removing the interior of an embedded
closed ball and identifying opposite hemispheres of its boundary. Consequently,
M 0 :D TOP.Dni f D; D¾i g/ is homeomorphic to M :D TOP.D/. This kind of
operation has been studied in the theory of GEMs, where it is called a fusion, see [LM].
3. If D6I ¡J D D¾i 6I ¡J holds, then M 0 is in general not homeomorphic to M .
The effect of the i -reduction varies. In dimension 2 TOP.Dni fD; D¾i g/ is derived
from M by cutting along an embedded circle and capping off the two boundaries
In dimension 3 there are four subcases depending on the set K :D f; 6D K ½
I ¡ J j D6K 6D D¾i 6K g. We define a Graph G D G.K/ :D .K; V .K// where
V .K/ :D ff A; Bg 2 K £ K j A ½ Bg. Since each K 2 K has either one or two
elements, each connected component of G is either a chain or a cycle. Let m be
the number of connected components.
(a) If m D 1 and G is a cycle, necessarily # J D 1 and the intersection S of the
simplices jDj and jD¾i j corresponding to D and D¾i in M consists of their
common i -face and the opposite vertex. Consequently, M 0 is derived from
M by removing a properly embedded solid torus and gluing in a cone on its
boundary. We call this operation a torus capping for the moment.
(b) If m D 1 and G is a chain, M 0 is homeomorphic to M .
(c) If m D 2, then M 0 is derived from M by cutting along a properly embedded
2sphere contained in jDj[jD¾i j and capping off the resulting boundary spheres
by disks. We call this a cut. If the result M 0 has two connected components,
then M is homeomorphic to the connected sum of the components of M 0 and
we call the operation a decomposition. Otherwise, we can express M as the
connected sum of M 0 with S2 £ S1. The operation of removing an S2 £ S1
summand is called a three-dimensional handle reduction.
(d) The case m D 3 can be described as a sequence of two cuts, each of which is
either a decomposition or a handle reduction.
We call an elementary i -reduction on a three-dimensional 6-set admissible if it is not
of type 3(a), i.e., if it is not a torus capping and its result is not empty. We call a general
i -reduction admissible if it can be decomposed into a series of admissible elementary
The following are immediate consequences:
Theorem 17. If D is a transitive, oriented, locally flat three-dimensional 6-set with at
least four elements and D0 is derived from D by a sequence of admissible elementary i
reductions, then TOP.D/ can be uniquely reconstructed given TOP.D0/ and the number
z of handle reductions performed, namely as the connected sum of all the connected
components of TOP.D0/ and z copies of S2 £ S1.
Corollary 18. Given D, D0 and z as above, TOP.D/ is a 3-torus if and only if exactly
one of the connected components of TOP.D0/ is a 3-torus, all others are 3-spheres and
z D 0.
Proof. This is obvious, because the 3-torus is prime.
Corollary 19. If D is a three-dimensional 6-set on which a handle-reduction can be
performed, then TOP.D/ is not a 3-torus.
Corollary 20. If D is an oriented, locally flat three-dimensional 6-set with at least four
elements and D0 is derived from D by a sequence of admissible elementary i -reductions,
then the number of handle reductions in any other sequence of admissible elementary
i -reductions which produce D0 is the same.
Moreover, we have
Lemma 21. If D0 D D¾i D D¾j holds for some element D of a three-dimensional
6-set D and i 6D j , then Dni fD; D¾i g is an admissible reduction.
It was shown in [FG] that a pair of GEMs of any dimension d represent homeomorphic
manifolds if and only if one can be transformed into the other by a sequence of fusions
and their inverses. This result applies directly to unwound, loopless, locally flat 6-sets.
Unfortunately, no general method is known to compute upper bounds for the minimal
lengths of such sequences. The isomorphism problem for manifolds can therefore not
be solved by systematically applying all sequences of fusions and inverse fusions on a
Anyway, it is straightforward to detect admissible elementary i -reductions in a 6-set,
construct the reduced symbol and—in dimension 3—determine the number of S2 £ S1
summands which were removed in the process. This process can then be repeated until
there are no further admissible i -reductions. However, for a practically useful algorithm,
such a strategy is not sufficient.
We now restrict our attention to dimension 3 once again.
A starting point for Lins’ approach [Li] is the observation that certain operations which
do not change the size of the 6-set may lead to a situation where an elementary reduction
can be applied. One such operation is the so-called p-pair switching: given D; D0 2 D
such that D0 2= fD; D¾i g but D0 2 D6i; j \ D6i;k for some triplet fi; j; kg ½ I , where
D is oriented and d.D/ 6D d.D0/, the ¾i -operation can be changed to connect D with D0
and the former D¾i with the former D0¾i . If D is locally flat, this increases the number
of 6i; j;k -orbits, thus allowing for an admissible l-reduction in the new symbol, where
fi; j; k; lg D I .
Another set of operations deals with configurations of three or more quadrangles.
A quadrangle in a 6-set is a loopless 6i; j -orbit of size 4 for some i; j 2 I . Certain
assemblies of quadrangles allow for local changes, so-called TS-moves, which do not
affect the topology. Lins’ algorithm constructs the (finite) list of all 6-sets which can
be derived from a given one, say D, by applying all possible sequences of TS-moves
and simplifies each of these as far as possible by sequences of elementary i -reduction.
If any of the results is smaller than D, the simplification process continues with it
instead of D. This is called the TS-algorithm. The reason for its usefulness lies in the
fact that whenever four quadrangles contain a common element, the 6-set in question
can be simplified. Moreover, by standard arguments based on the Euler characteristic, a
loopless, locally flat, three-dimensional 6-set without 6i; j -orbits of size 2 must contain
at least 12 quadrangles.
Going even further, Lins introduces an operation, called the U -move, which increases
the size of a 6-set, but introduces many quadrangles. Depending on the number k of
U -moves which are allowed before applying the TS-algorithm, this gives rises to an
infinite number of so-called Uk -algorithms.
In contrast, D-symbols and 6-sets derived from tilings have a special property which
we call regularity. A d-dimensional D-symbol .D; M / is regular if mi j .D/ D 2 holds
for all D 2 D and all pairs of indices satisfying ji ¡ j j > 1. A d-dimensional 6-set D
is regular if for indices i; j satisfying ji ¡ j j > 1, every 6i; j -orbit is a quadrangle. The
D-symbol of a tiling is always regular. If .D; M / is regular, unwound and loopless, then
D is regular.
The simplification process we use in our applications to tiling theory is designed
to retain—and make use of—regularity in each step. The basic operation is a
nonelementary i -reduction which we call a collapse. Consider fi; j; kg ½ I and D 2 D such
that every 6i; j - and 6i;k -orbit in D6i; j;k is a quadrangle. In other words, D0.¾i ¾j /2 D
D0.¾i ¾k /2 D D0 for every D0 2 D6i; j;k . If D is oriented, Dni fD; D¾i g is admissible
and D ¡ D6i; j;k is non-empty, then Dni .D6i; j;k / is admissible. The proof is a simple
induction based on Lemma 21. We call Dni .D6i; j;k / an .i I j; k/-collapse. Obviously,
collapses preserve regularity.
Any admissible 0-reduction on a regular, oriented 6-set leads to an admissible
.0I 2; 3/-collapse. Likewise, any admissible 3-reduction leads to an admissible .3I 0;
1/collapse. These two types of collapses can be interpreted as the removal of walls (2-faces)
and the shrinking of edges in the corresponding tiling. Sometimes 1- or 2-reductions
lead to admissible collapses as well, which can be interpreted as the removal of
certain degeneracies in the tiling. As in Lins’ algorithm, we make use of a limited set
of operations which do not decrease the size of the 6-set. These are expressed as
combinations of collapses and their inverses. Unlike Lins, we are able to introduce a
slightly more complicated complexity measurement such that each step of our
algorithm decreases the complexity of the 6-set, thus by showing that this complexity can
only be decreased a finite number of times we establish that the algorithm eventually
Details will appear in a forthcoming paper, in which we will also address reduction
methods for general, i.e., possibly not unwound, D-symbols.
9. An Example
To illustrate the ideas of this text, we consider the classification of flat regular D-symbols
with a common underlying 60;1;2;3-set D0 shown in Fig. 1. In our figures the ¾i are
represented by edges of different styles, as indicated in the legend in Fig. 1.
thus K > 0 implies 2=c C 4=d > 1. We discard solutions with c or d smaller than 3,
as these would correspond to “degenerate” tilings with certain edges incident to only
one or two tiles. Furthermore, by the crystallographic restriction, in a flat D-symbol we
must have vi; j .D/ 2 f1; 2; 3; 4; 6g for all i; j 2 I and D 2 D. There remain six possible
combinations of values for c and d, namely .3; 4/, .3; 6/, .3; 8/, .4; 4/, .4; 6/ and .6; 4/.
Each of these is admissible in the sense that it induces a spherical 61;2;3-subsymbol. The
corresponding complete D-symbols will be denoted by .D; M3;4/, .D; M3;6/ and so on.
From the combination of branching numbers in each individual two-dimensional
subsymbol, one can determine the geometric type of a corresponding vertex-, edge-,
faceor tile-stabilizer in a flat realization. As each of these stabilizers must be contained in
the point group, we may find that certain point groups are impossible. Only
orientationpreserving groups have to be considered, as D is oriented. Here, it turns out that for
the cases .3; 6/, .3; 8/ and .4; 6/, the point group must be the group of
orientationpreserving symmetries of the octahedron, labelled O in Scho¨nfliess notation and 432 in
crystallographic notation. This group is abstractly isomorphic to the symmetric group
S4. In the cases .3; 8/ and .4; 6/, this is obvious from the fact that there are rotations
of both orders 3 and 4, whereas in case .3; 6/, both the groups T (or 23) and D3 (or
322) occur as stabilizers, which are conjugate to common subgroups only in 432. In the
case .6; 4/, the point group must clearly be D6 (or 622), whereas in the case .4; 4/, the
point group cannot be one of T or O due to the fact that a three-dimensional irreducible
crystallographic space group must contain a rotation of order 3, which leaves D4 (or
422) as the only possible point group. Only in the case .3; 4/ is the stabilizer information
not sufficient to uniquely determine the geometric type of the point group.
Next we check whether any of the six fundamental groups in question contains a
normal subgroup with factor group isomorphic to S4. We start by enumerating arbitrary
homomorphisms into S4, i.e., actions of these groups on a four-element set and then pick
out the surjective ones. Our computer implementation constructs explicit presentations
of the fundamental groups in terms of abstract generators and relators. It is usually more
efficient to work on these, but in view of Lemma 9, we can equivalently construct all
transitive fourfold coverings of the given D-symbol and read off the required information
We assign labels from the set f A; B; C; Dg £ D to the elements of the covering
Dsymbol. These are abbreviated to A1, A2; : : : in the figures. The first component of each
label corresponds to an element of the set f A; B; C; Dg, while the second component
determines the image in the 6-set D. Given a 6-map onto D which matches these
labels, we can assign to each pair .D; i / 2 D £ f0; 1; 2; 3g a permution ® of the set
f A; B; C; Dg by defining ®.X / :D Y precisely if .X; D/¾i D .Y; D¾i / holds. The pairs
.D; i / can be identified with generators of the fundamental group G :D ¼1.D; M /. The
corresponding map G ! S4 is well-defined if and only if the 6-map induces a covering
of D-symbols. This methods works because the fundamental group of the cover will be
just the stabilizer of some element with respect to the action on f A; B; C; Dg.
If ': .D0; M 0/ ! .D; M / is a covering, then its restriction to any 6J -subsymbol must
induce a covering of 6J -symbols. We therefore start by looking at possible two-, three- or
fourfold covers of the 60;1;2-subsymbol common to all our candidates. Because m0;1 ´
m1;2 ´ 3, both r0;1.X; D/ and r1;2.X; D/ must be 1 or 3 for each X 2 f A; B; C; Dg and
D 2 D. There are essentially only two solutions, as shown in Figs. 3 and 4. Thus, each
fourfold cover of a D-symbol in our list either contains exactly one 60;1;2-subsymbol,
which must be isomorphic to the one in Fig. 3, or exactly two 60;1;2-subsymbols, which
must both be isomorphic to the one in Fig. 4.
The D-symbol cover corresponding to Fig. 3 is obviously unwound. Incidentally, it
is even universal. We read off the permutations ® :D . A; D/.B; C / for the pair .1; 2/,
¯ :D . A; B/.C; D/ for .3; 2/ and ° :D . A; C /.B; D/ for .5; 2/. From the remaining
pairs, we obtain nothing new. Our next task is to determine all possible ¾3-actions which
induce permutations that together with ®, ¯ and ° generate S4. However, by the regularity
condition and the structure of D, the ¾3-action can only lead to one new permutation ±,
which must be an involution. Thus ± must be one of ®, ¯ and ° , which together generate
Fig. 3. The universal cover of D.1;f0;1;2g/.
a four-element group, or a transposition, in which case the generated group has eight
elements. Note that this fact is independent of the actual values of c and d.
It remains to consider the case where the 6-preimage of D contains two 60;1;2-orbits,
each of which is isomorphic to the 6-set shown in Fig. 4. There are three essentially
different labellings for this set, according to whether the 60;2-orbits of size 2 are mapped
onto f1; 2g, f3; 4g or f5; 6g, respectively. In any case, we read off no non-trivial
permutations from the ¾0- and ¾1-actions and just one from the ¾2-action, which by appropriate
labelling can be assumed to be . A; B/.C; D/. Again, we obtain at most one more
permutation from the ¾3-action, which must be of order 2. In order to generate S4 together
with . A; B/.C; D/, this must be a transposition other than . A; B/ and .C; D/. Again by
appropriate labelling, we may assume that this transposition is .B; C /.
Now by constructing the unique 6-set corresponding to each of the nine possible
combinations of the second parts of the labels and eliminating isomorphic copies, one
easily checks that there are exactly four 6-preimages of D to be considered, namely those
depicted in Figs. 5–8. For each of these and each of our six candidate D-symbols, we can
now check whether the 6-map onto D extends to a D-symbol covering, i.e., whether the
r2;3-values which can be read off from the 6-set divide the corresponding m2;3-values
induced by the map. It turns out thus that Case I is compatible with symbols .D; M4;4/
and .D; M6;4/ while Case II is compatible only with symbol .D; M4;4/, Case III with
.D; M3;8/ and Case IV with .D; M4;6/.
Given a D-symbol .D; M /, a fourfold cover .D0; M 0/ and a corresponding
homomorphism ': ¼1.D; M / ! S4, our next task is to construct the cover of .D; M / which
corresponds to the kernel of that homomorphism. The following is a very convenient
way to do that: to each pair .D; i / 2 D £ I , we have associated a permutation of
f A; B; C; Dg. Now we define a new 6-set DO D S4 £ D with the 6-action given by
.¯; D/¾i :D .¯ ± ®; D¾i /, where ® is the permutation associated to .D; i /. That is,
we construct the cover corresponding to a stabilizer with respect to the multiplication
action of ¼1.D; M / on S4 induced by the homomorphism ', which of course is
exactly the kernel of '. The neat thing about this construction is that we do not actually
have to construct the 6-set DO and the induced cover .DO ; MO / to see whether the latter
is unwound. For example, in Case I (Fig. 5) of our list of covers, the trivial
permutation is associated to each of the pairs .1; 0/, .1; 3/, .2; 0/ and .2; 3/, which implies
r0;3.¯; 1/ D 1 and thus by regularity v0;3.¯; 1/ D 2 for every ¯ 2 S4. Similarly,
in Case II (Fig. 6), the trivial permutation is associated to each of the pairs .3; 0/,
.3; 3/, .4; 0/ and .4; 3/. Consequently, .DO ; MO / is not unwound for any of these two
Incidentally, we had already noted that the symbols .D; M4;4/ and .D; M6;4/ cannot
correspond to tilings with point group O, which gives us a second independent reason
to reject Cases I and II.
In Case III (Fig. 7), the cover .DO ; MO / turns out to be unwound, but the commutator
factor group of its fundamental group is Z2 instead of Z3. Indeed, the corresponding
D-symbol .D; M3;8/ encodes a tiling of the 3-sphere by 288 tetrahedra.
In Case IV (Fig. 8), the cover is also unwound, and in this single case all further tests
are successful. Thus, the D-symbol .D; M6;4/ encodes a tiling of E3. Testing for other
possible point groups where necessary does not reveal any other flat symbols, so we may
conclude that there is exactly one flat, regular non-degenerate D-symbol with underlying
As a final remark, we note that in this particular case, it is easy to check the results
geometrically. From the structure of the universal cover of the 60;1;2-subsymbol of D0,
shown in Fig. 3, we can deduce that a tiling based on D0 must have combinatorially
tetrahedral tiles. Since ¾3 acts trivially, each tile is mapped onto each of its neighbors
by a mirror reflection. Thus, all 2-faces must be flat and the tiles are actually convex
tetrahedra with their vertices as extremal points. Furthermore, they possess two opposite
edges with dihedral angles of 2¼=c, while all other edges have dihedral angles of 2¼=d.
In a euclidean tetrahedron, this is only possible for c D 4 and d D 6.
Fig. 7. Case III.
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